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January 29, 2020

 Calculating the limit of the function   I.S.I. (Indian Statistical Institute) B.Stat/B.Math Entrance Examination 2016, problem 21

[et_pb_section fb_built="1" _builder_version="3.22.4" custom_margin="-1px|||||"][et_pb_row _builder_version="4.1"][et_pb_column type="4_4" _builder_version="3.25" custom_padding="|||" custom_padding__hover="|||"][et_pb_text _builder_version="4.1" text_font="Raleway|300|||||||" text_text_color="#ffffff" header_font="Raleway|300|||||||" header_text_color="#0c71c3" background_color="#0c71c3" transform_scale="5%|5%" custom_padding="20px|20px|20px|20px" border_radii="on|5px|5px|5px|5px" box_shadow_style="preset3"]

What are we learning ?

[/et_pb_text][et_pb_text _builder_version="4.1" text_font="Raleway||||||||" background_color="#f4f4f4" custom_margin="10px||10px" custom_padding="10px|20px|10px|20px" box_shadow_style="preset2"]Competency in Focus: Calculating the limit of the function  This problem from I.S.I. (Indian Statistical Institute) B.Stat/B.Math Entrance Examination 2016 is based on simple manipulations and limit of a function .[/et_pb_text][et_pb_text _builder_version="4.1" text_text_color="#ffffff" header_font="Raleway|300|||||||" header_text_color="#ffffff" background_color="#0c71c3" width="100%" custom_margin="48px||48px||false|false" custom_padding="20px|20px|20px|20px|false|false" border_radii="on|5px|5px|5px|5px" border_color_all="#0c71c3"]

First look at the knowledge graph.

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Next understand the problem

[/et_pb_text][et_pb_text _builder_version="4.1" hover_enabled="0" border_radii="on|100px|100px|100px|100px" box_shadow_style="preset3" box_shadow_color="#ffffff"]Let \(f:R \to R\) be a non-zero function such that \(\lim_{x \rightarrow \infty} \frac{f(xy)}{x^3}\) exists for all y >0. Let \(g(y)= \lim_{x \rightarrow \infty} \frac{f(xy)}{x^3}\) .If g(1)=1 ,then for all y>0 ? [/et_pb_text][/et_pb_column][/et_pb_row][et_pb_row _builder_version="3.25"][et_pb_column type="4_4" _builder_version="3.25" custom_padding="|||" custom_padding__hover="|||"][et_pb_accordion open_toggle_text_color="#0c71c3" _builder_version="4.1" toggle_font="||||||||" body_font="Raleway||||||||" text_orientation="center" custom_margin="10px||10px"][et_pb_accordion_item title="Source of the problem" open="on" _builder_version="4.1"]
I.S.I. (Indian Statistical Institute) B.Stat/B.Math Entrance Examination 2016. Obective Problem no. 21.
[/et_pb_accordion_item][et_pb_accordion_item title="Key Competency" _builder_version="4.1" open="off"]Limit of a function [/et_pb_accordion_item][et_pb_accordion_item title="Difficulty Level" _builder_version="4.1" open="off"]7 out of 10[/et_pb_accordion_item][et_pb_accordion_item title="Suggested Book" _builder_version="3.22.4" open="off"]Elementary Number Theory by David M. Burton

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Start with hints

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So , what we have to find in general the value of  g(y) in terms of y  given that \(g(y)=\lim_{x \rightarrow \infty} \frac{f(xy)}{x^3}\) provided \(g(y)=\lim_{x \rightarrow \infty} \frac{f(xy)}{x^3}\) exists .[/et_pb_tab][et_pb_tab title="Hint 2" _builder_version="4.1" hover_enabled="0"]

  See what can we do is that g(y) can be written as follows , \(g(y)=\lim_{x \rightarrow \infty} \frac{f(xy)}{x^3}=
(y^3)\lim_{x \rightarrow \infty} \frac{f(xy)}{x^3y^3} \) [/et_pb_tab][et_pb_tab title="Hint 3" _builder_version="4.1" hover_enabled="0"]  Again from the given condition we have \(g(1)=\lim_{x \rightarrow \infty} f(x) /x^3=\lim_{xy \rightarrow \infty} \frac{f(xy)}{x^3y^3}=1\).[/et_pb_tab][et_pb_tab title="Hint 4" _builder_version="4.1" hover_enabled="0"]Therefore , \(g(y)=(y^3)\lim_{x \rightarrow \infty} \frac{f(xy)}{x^3y^3} =y^3 \) by previous argument .  [/et_pb_tab][/et_pb_tabs][et_pb_text _builder_version="3.27.4" text_font="Raleway|300|||||||" text_text_color="#ffffff" header_font="Raleway|300|||||||" header_text_color="#e2e2e2" background_color="#0c71c3" min_height="12px" custom_margin="50px||50px" custom_padding="20px|20px|20px|20px" border_radii="on|5px|5px|5px|5px" box_shadow_style="preset3"]

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Similar Problem

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